Molecular Quantum Mechanics

by Atkins, Peter; Friedman, Ronald

• ISBN13: 9780199274987
• ISBN10: 0199274983
• Edition: 4th
• Format: Paperback
• Copyright: 2005-02-17
• Publisher: Oxford University Press

Summary

Quantum mechanics embraces the behaviour of all known forms of matter, including the atoms and molecules from which we, and all living organisms, are composed. Molecular Quantum Mechanics leads us through this absorbing yet challenging subject, unravelling those fundamental physical principleswhich explain how all matter behaves.With the clarity of exposition and rich pedagogy which have established the book as a leading text in the field, Molecular Quantum Mechanics takes us from the foundations of quantum mechanics, through quantum models of atomic, molecular, and electronic structure, and on to discussions ofspectroscopy, and the electronic and magnetic properties of molecules. Lucid explanations and illuminating artworks help to visualise the many abstract concepts upon which the subject is built.Fully updated to reflect the latest advances in computational techniques, and enhanced with more mathematical support and worked examples than ever before, Molecular Quantum Mechanics remains the ultimate resource for those wishing to master this important subject.Supplementary resourcesOnline Resource Centre, featuring:Illustrations available to downloadSolutions manual available to download [instructors only]

Table of Contents

Introduction and orientation	1	(1)
Black-body radiation	1	(2)
Heat capacities	3	(1)
The photoelectric and Compton effects	4	(1)
Atomic spectra	5	(1)
The duality of matter	6	(2)
Problems	8	(1)
The foundations of quantum mechanics	9	(34)
Operators in quantum mechanics	9	(10)
Linear operators	10	(1)
Eigenfunctions and eigenvalues	10	(2)
Representations	12	(1)
Commutation and non-commutation	13	(1)
The construction of operators	14	(1)
Integrals over operators	15	(1)
Dirac bracket notation	16	(1)
Hermitian operators	17	(2)
The postulates of quantum mechanics	19	(6)
States and wavefunctions	19	(1)
The fundamental prescription	20	(1)
The outcome of measurements	20	(2)
The interpretation of the wavefunction	22	(1)
The equation for the wavefunction	23	(1)
The separation of the Schrodinger equation	23	(2)
The specification and evolution of states	25	(7)
Simultaneous observables	25	(2)
The uncertainty principle	27	(2)
Consequences of the uncertainty principle	29	(1)
The uncertainty in energy and time	30	(1)
Time-evolution and conservation laws	30	(2)
Matrices in quantum mechanics	32	(4)
Matrix elements	32	(2)
The diagonalization of the hamiltonian	34	(2)
The plausibility of the Schrodinger equation	36	(7)
The propagation of light	36	(2)
The propagation of particles	38	(1)
The transition to quantum mechanics	39	(1)
Problems	40	(3)
Linear motion and the harmonic oscillator	43	(28)
The characteristics of acceptable wavefunctions	43	(1)
Some general remarks on the Schrodinger equation	44	(3)
The curvature of the wavefunction	45	(1)
Qualitative solutions	45	(1)
The emergence of quantization	46	(1)
Penetration into non-classical regions	46	(1)
Translational motion	47	(4)
Energy and momentum	48	(1)
The significance of the coefficients	48	(1)
The flux density	49	(1)
Wavepackets	50	(1)
Penetration into and through barriers	51	(4)
An infinitely thick potential wall	51	(1)
A barrier of finite width	52	(2)
The Eckart potential barrier	54	(1)
Particle in a box	55	(5)
The solutions	56	(1)
Features of the solutions	57	(1)
The two-dimensional square well	58	(1)
Degeneracy	59	(1)
The harmonic oscillator	60	(6)
The solutions	61	(2)
Properties of the solutions	63	(2)
The classical limit	65	(1)
Translation revisited: The scattering matrix	66	(5)
Problems	68	(3)
Rotational motion and the hydrogen atom	71	(27)
Particle on a ring	71	(5)
The hamiltonian and the Schrodinger equation	71	(2)
The angular momentum	73	(1)
The shapes of the wavefunctions	74	(2)
The classical limit	76	(1)
Particle on a sphere	76	(8)
The Schrodinger equation and its solution	76	(3)
The angular momentum of the particle	79	(2)
Properties of the solutions	81	(1)
The rigid rotor	82	(2)
Motion in a Coulombic field	84	(14)
The Schrodinger equation for hydrogenic atoms	84	(1)
The separation of the relative coordinates	85	(1)
The radial Schrodinger equation	85	(5)
Probabilities and the radial distribution function	90	(1)
Atomic orbitals	91	(3)
The degeneracy of hydrogenic atoms	94	(2)
Problems	96	(2)
Angular momentum	98	(24)
The angular momentum operators	98	(4)
The operators and their commutation relations	99	(2)
Angular momentum observables	101	(1)
The shift operators	101	(1)
The definition of the states	102	(10)
The effect of the shift operators	102	(2)
The eigenvalues of the angular momentum	104	(2)
The matrix elements of the angular momentum	106	(2)
The angular momentum eigenfunctions	108	(2)
Spin	110	(2)
The angular momenta of composite systems	112	(10)
The specification of coupled states	112	(1)
The permitted values of the total angular momentum	113	(2)
The vector model of coupled angular momenta	115	(2)
The relation between schemes	117	(2)
The coupling of several angular momenta	119	(1)
Problems	120	(2)
Group theory	122	(46)
The symmetries of objects	122	(7)
Symmetry operations and elements	123	(1)
The classification of molecules	124	(5)
The calculus of symmetry	129	(16)
The definition of a group	129	(1)
Group multiplication tables	130	(1)
Matrix representations	131	(4)
The properties of matrix representations	135	(2)
The characters of representations	137	(1)
Characters and classes	138	(1)
Irreducible representations	139	(3)
The great and little orthogonality theorems	142	(3)
Reduced representations	145	(6)
The reduction of representations	146	(1)
Symmetry-adapted bases	147	(4)
The symmetry properties of functions	151	(10)
The transformation of p-orbitals	151	(1)
The decomposition of direct-product bases	152	(3)
Direct-product groups	155	(2)
Vanishing integrals	157	(2)
Symmetry and degeneracy	159	(2)
The full rotation group	161	(4)
The generators of rotations	161	(1)
The representation of the full rotation group	162	(2)
Coupled angular momenta	164	(1)
Applications	165	(3)
Problems	166	(2)
Techniques of approximation	168	(39)
Time-independent perturbation theory	168	(15)
Perturbation of a two-level system	169	(2)
Many-level systems	171	(1)
The first-order correction to the energy	172	(2)
The first-order correction to the wavefunction	174	(1)
The second-order correction to the energy	175	(1)
Comments on the perturbation expressions	176	(2)
The closure approximation	178	(2)
Perturbation theory for degenerate states	180	(3)
Variation theory	183	(4)
The Rayleigh ratio	183	(2)
The Rayleigh-Ritz method	185	(2)
The Hellmann-Feynman theorem	187	(2)
Time-dependent perturbation theory	189	(18)
The time-dependent behaviour of a two-level system	189	(3)
The Rabi formula	192	(1)
Many-level systems: the variation of constants	193	(2)
The effect of a slowly switched constant perturbation	195	(2)
The effect of an oscillating perturbation	197	(2)
Transition rates to continuum states	199	(1)
The Einstein transition probabilities	200	(3)
Lifetime and energy uncertainty	203	(1)
Problems	204	(3)
Atomic spectra and atomic structure	207	(42)
The spectrum of atomic hydrogen	207	(12)
The energies of the transitions	208	(1)
Selection rules	209	(3)
Orbital and spin magnetic moments	212	(2)
Spin-orbit coupling	214	(2)
The fine-structure of spectra	216	(1)
Term symbols and spectral details	217	(1)
The detailed spectrum of hydrogen	218	(1)
The structure of helium	219	(10)
The helium atom	219	(3)
Excited states of helium	222	(2)
The spectrum of helium	224	(1)
The Pauli principle	225	(4)
Many-electron atoms	229	(13)
Penetration and shielding	229	(2)
Periodicity	231	(2)
Slater atomic orbitals	233	(1)
Self-consistent fields	234	(2)
Term symbols and transitions of many-electron atoms	236	(3)
Hund's rules and the relative energies of terms	239	(1)
Alternative coupling schemes	240	(2)
Atoms in external fields	242	(7)
The normal Zeeman effect	242	(1)
The anomalous Zeeman effect	243	(2)
The Stark effect	245	(1)
Problems	246	(3)
An introduction to molecular structure	249	(38)
The Born-Oppenheimer approximation	249	(4)
The formulation of the approximation	250	(1)
An application: the hydrogen molecule-ion	251	(2)
Molecular orbital theory	253	(13)
Linear combinations of atomic orbitals	253	(5)
The hydrogen molecule	258	(1)
Configuration interaction	259	(2)
Diatomic molecules	261	(4)
Heteronuclear diatomic molecules	265	(1)
Molecular orbital theory of polyatomic molecules	266	(12)
Symmetry-adapted linear combinations	266	(3)
Conjugated π-systems	269	(5)
Ligand field theory	274	(2)
Further aspects of ligand field theory	276	(2)
The band theory of solids	278	(9)
The tight-binding approximation	279	(2)
The Kronig-Penney model	281	(3)
Brillouin zones	284	(1)
Problems	285	(2)
The calculation of electronic structure	287	(55)
The Hartree-Fock self-consistent field method	288	(14)
The formulation of the approach	288	(1)
The Hartree-Fock approach	289	(2)
Restricted and unrestricted Hartree-Fock calculations	291	(2)
The Roothaan equations	293	(3)
The selection of basis sets	296	(5)
Calculational accuracy and the basis set	301	(1)
Electron correlation	302	(14)
Configuration state functions	303	(1)
Configuration interaction	303	(2)
Cl calculations	305	(3)
Multiconfiguration and multireference methods	308	(2)
Moller-Plesset many-body perturbation theory	310	(3)
The coupled-cluster method	313	(3)
Density functional theory	316	(5)
Kohn-Sham orbitals and equations	317	(2)
Exchange-correlation functionals	319	(2)
Gradient methods and molecular properties	321	(4)
Energy derivatives and the Hessian matrix	321	(1)
Analytical derivatives and the coupled perturbed equations	322	(3)
Semiempirical methods	325	(7)
Conjugated π-electron systems	326	(3)
Neglect of differential overlap	329	(3)
Molecular mechanics	332	(4)
Force fields	333	(1)
Quantum mechanics-molecular mechanics	334	(2)
Software packages for electronic structure calculations	336	(6)
Problems	339	(3)
Molecular rotations and vibrations	342	(40)
Spectroscopic transitions	342	(2)
Absorption and emission	342	(2)
Raman processes	344	(1)
Molecular rotation	344	(13)
Rotational energy levels	345	(4)
Centrifugal distortion	349	(1)
Pure rotational selection rules	349	(2)
Rotational Raman selection rules	351	(2)
Nuclear statistics	353	(4)
The vibrations of diatomic molecules	357	(8)
The vibrational energy levels of diatomic molecules	357	(2)
Anharmonic oscillation	359	(1)
Vibrational selection rules	360	(2)
Vibration-rotation spectra of diatomic molecules	362	(2)
Vibrational Raman transitions of diatomic molecules	364	(1)
The vibrations of polyatomic molecules	365	(14)
Normal modes	365	(3)
Vibrational selection rules for polyatomic molecules	368	(1)
Group theory and molecular vibrations	369	(4)
The effects of anharmonicity	373	(3)
Coriolis forces	376	(1)
Inversion doubling	377	(2)
Appendix 10.1 Centrifugal distortion	379	(3)
Problems	380	(2)
Molecular electronic transitions	382	(25)
The states of diatomic molecules	382	(4)
The Hund coupling cases	382	(2)
Decoupling and A-doubling	384	(2)
Selection rules	386	(1)
Vibronic transitions	386	(4)
The Franck-Condon principle	386	(3)
The rotational structure of vibronic transitions	389	(1)
The electronic spectra of polyatomic molecules	390	(6)
Symmetry considerations	391	(1)
Chromophores	391	(2)
Vibronically allowed transitions	393	(2)
Singlet-triplet transitions	395	(1)
The fate of excited species	396	(11)
Non-radiative decay	396	(1)
Radiative decay	397	(2)
The conservation of orbital symmetry	399	(1)
Electrocyclic reactions	399	(2)
Cycloaddition reactions	401	(2)
Photochemically induced electrocyclic reactions	403	(1)
Photochemically induced cycloaddition reactions	404	(2)
Problems	406	(1)
The electric properties of molecules	407	(29)
The response to electric fields	407	(11)
Molecular response parameters	407	(2)
The static electric polarizability	409	(2)
Polarizability and molecular properties	411	(2)
Polarizabilities and molecular spectroscopy	413	(1)
Polarizabilities and dispersion forces	414	(4)
Retardation effects	418	(1)
Bulk electrical properties	418	(9)
The relative permittivity and the electric susceptibility	418	(2)
Polar molecules	420	(2)
Refractive index	422	(5)
Optical activity	427	(9)
Circular birefringence and optical rotation	427	(2)
Magnetically induced polarization	429	(2)
Rotational strength	431	(3)
Problems	434	(2)
The magnetic properties of molecules	436	(37)
The descriptions of magnetic fields	436	(6)
The magnetic susceptibility	436	(1)
Paramagnetism	437	(2)
Vector functions	439	(1)
Derivatives of vector functions	440	(1)
The vector potential	441	(1)
Magnetic perturbations	442	(10)
The perturbation hamiltonian	442	(2)
The magnetic susceptibility	444	(3)
The current density	447	(3)
The diamagnetic current density	450	(1)
The paramagnetic current density	451	(1)
Magnetic resonance parameters	452	(21)
Shielding constants	452	(4)
The diamagnetic contribution to shielding	456	(2)
The paramagnetic contribution to shielding	458	(1)
The g-value	459	(3)
Spin-spin coupling	462	(1)
Hyperfine interactions	463	(4)
Nuclear spin-spin coupling	467	(4)
Problems	471	(2)
Scattering theory	473	(40)
The formulation of scattering events	473	(6)
The scattering cross-section	473	(2)
Stationary scattering states	475	(4)
Partial-wave stationary scattering states	479	(18)
Partial waves	479	(1)
The partial-wave equation	480	(1)
Free-particle radial wavefunctions and the scattering phase shift	481	(3)
The JWKB approximation and phase shifts	484	(2)
Phase shifts and the scattering matrix element	486	(2)
Phase shifts and scattering cross-sections	488	(2)
Scattering by a spherical square well	490	(2)
Background and resonance phase shifts	492	(2)
The Breit-Wigner formula	494	(1)
Resonance contributions to the scattering matrix element	495	(2)
Multichannel scattering	497	(5)
Channels for scattering	497	(1)
Multichannel stationary scattering states	498	(1)
Inelastic collisions	498	(3)
The S matrix and multichannel resonances	501	(1)
The Green's function	502	(6)
The integral scattering equation and Green's functions	502	(2)
The Born approximation	504	(4)
Appendix 14.1 The derivation of the Breit-Wigner formula	508	(1)
Appendix 14.2 The rate constant for reactive scattering	509	(4)
Problems	510	(3)
Further information	513	(40)
Classical mechanics	513	(6)
Action	513	(2)
The canonical momentum	515	(1)
The virial theorem	516	(2)
Reduced mass	518	(1)
Solutions of the Schrodinger equation	519	(15)
The motion of wavepackets	519	(2)
The harmonic oscillator: solution by factorization	521	(2)
The harmonic oscillator: the standard solution	523	(2)
The radial wave equation	525	(1)
The angular wavefunction	526	(1)
Molecular integrals	527	(1)
The Hartree--Fock equations	528	(4)
Green's functions	532	(1)
The unitarity of the S matrix	533	(1)
Group theory and angular momentum	534	(3)
The orthogonality of basis functions	534	(1)
Vector coupling coefficients	535	(2)
Spectroscopic properties	537	(6)
Electric dipole transitions	537	(1)
Oscillator strength	538	(2)
Sum rules	540	(1)
Normal modes: an example	541	(2)
The electromagnetic field	543	(4)
The Maxwell equations	543	(3)
The dipolar vector potential	546	(1)
Mathematical relations	547	(6)
Vector properties	547	(2)
Matrices	549	(4)
Further reading	553	(4)
Appendix 1	557	(5)
Appendix 2	562	(1)
Answers to selected problems	563	(2)
Index	565